The Stacks project

Lemma 65.23.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

  1. $f$ is locally of finite type,

  2. for every $x \in |X|$ the morphism $f$ is of finite type at $x$,

  3. for every scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is locally of finite type,

  4. for every affine scheme $Z$ and any morphism $Z \to Y$ the morphism $Z \times _ Y X \to Z$ is locally of finite type,

  5. there exists a scheme $V$ and a surjective étale morphism $V \to Y$ such that $V \times _ Y X \to V$ is locally of finite type,

  6. there exists a scheme $U$ and a surjective étale morphism $\varphi : U \to X$ such that the composition $f \circ \varphi $ is locally of finite type,

  7. for every commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    where $U$, $V$ are schemes and the vertical arrows are étale the top horizontal arrow is locally of finite type,

  8. there exists a commutative diagram

    \[ \xymatrix{ U \ar[d] \ar[r] & V \ar[d] \\ X \ar[r] & Y } \]

    where $U$, $V$ are schemes, the vertical arrows are étale, $U \to X$ is surjective, and the top horizontal arrow is locally of finite type, and

  9. there exist Zariski coverings $Y = \bigcup _{i \in I} Y_ i$, and $f^{-1}(Y_ i) = \bigcup X_{ij}$ such that each morphism $X_{ij} \to Y_ i$ is locally of finite type.

Proof. Each of the conditions (2), (3), (4), (5), (6), (7), and (9) imply condition (8) in a straightforward manner. For example, if (5) holds, then we can choose a scheme $V$ which is a disjoint union of affines and a surjective morphism $V \to Y$ (see Properties of Spaces, Lemma 64.6.1). Then $V \times _ Y X \to V$ is locally of finite type by (5). Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to V$ is locally of finite type by Lemma 65.23.2. Hence (8) is true.

The conditions (1), (7), and (8) are equivalent by definition.

To finish the proof, we show that (1) implies all of the conditions (2), (3), (4), (5), (6), and (9). For (2) this is immediate. For (3), (4), (5), and (9) this follows from the fact that being locally of finite type is preserved under base change, see Lemma 65.23.3. For (6) we can take $U = X$ and we're done. $\square$


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